Shapes and Angles
The world is made of a combination of shapes and angles. Students learn about mathematical perfect shapes and angles, but in reality the terms were developed to help describe the real world. Students should look at the pictures of natural occurring shapes.
The plant has bumps which are nearly parallel; the spider web is composed of radiating bowed rectangles; the giraffe neck is around 110 degrees from its body; and the mountain is triangular in shape. You may want students to look around them and have force them to see shapes and angles.
Geometry is from the Greek word “geo” meaning earth and “metria” meaning measurement. Geometry basically measures components on Earth.
Math models the world. However, you can look at the world differently and you get different models. Geometry, for example, deals with deduction of properties, measurements, and relationships of points, lines, angles, and figures in space. Euclidean geometry deals with a flat world that states that parallel lines are always parallel and never intersect. Euclidean geometry is our way of measuring on Earth.
On the other hand, Riemannian geometry deals with a spherical world that says parallel lines will intersect. This takes into consideration, the Universe may be a closed surface.
Geometry analyzes shapes. Young children are brought into an exciting world of different shapes that can be seen and felt. Early toys allow a child to feel and begin recognizing shapes. However, as children enter school their 3-dimensional world becomes very flat due to the introduction of paper and pencil. The world of balls and cubes turns into circles and rectangles. It is not until high school do we ask children to analyze 3 dimensional shapes. That sometimes causes of problem in visualization of how the real world actually appears.
The next 6 slides reviews or introduces the different types of solids and their basic definitions. In the real world we only use solids; two-dimensional space is only for a flat paper world.
Spheres are solids without edges, making them difficult to measure. The same distance from a fixed pointed refers to the radius and the center. Ask students to visualize the entire same radius coming from the center. Ask students to give you examples of spheres from balls they play with to ball bearings.
A cylinder also has no edges. It is a set of the same or congruent shapes that are parallel to each other. A cylinder is basically a straight line with equidistant (radius) lines that radiate from the center line. A rolling bin is a cylinder,
The cone has a sharp point, but the rest of the solid is curved. The cone has a round base and it creates straight lines to the vertex or apex. Ask students to think identify cones in their everyday life from ice cream cones to traffic cones..
Polyhedron has edges, vertices (or corners), and faces. Polyhedron has no curved surfaces. The face refers to one side, while the entire polyhedron is referred to as the body. A polyhedron is a set of 4 or more polygons that enclose a three dimensional structure or solid. Each flat plane is called a face and has a common segment called an edge with various angles. The angle is measured in the plane perpendicular to the edge. Three or more edges form a vertex.
Polyhedrons are used in building because it is very stable. There are many different types of polyhedrons from a simple cube to a complex dodecahedron,
A prism has edges and is a specialized polyhedron in that two sides are congruent and parallel. The base of a prism is a polygon. A prism has a cross section along the entire length. A prism shape glass or plastic can refract white light into the colors of the rainbow. Note to students that an optical prism takes on the name of the shape is takes.
A pyramid has a base composed of a polygon and the lateral or side faces are always triangles. Pyramids come in regular, right and oblique varieties. A tetrahedron is an example of a regular pyramid.
The Platonic solids are composed of a single repeating polygon. There are only five Platonic solids including the tetrahedron (4 triangles), cube (6 squares), icosahedron (20 triangles), octahedron (8 triangles) and dodecahedron (12 pentagons). These are only 5 solids that are geometrically possible.
There is evidence that the Egyptians knew about at least three of the solids; their work influenced the Pythagoreans. In any case, Plato mentioned these solids in his writings He identified them with the elements then commonly believed to make up all matter in the universe; fire, air, water, earth, and the cosmos (the universe itself). Plato identified fire atoms with the tetrahedron, earth atoms with the cube, air atoms with the octahedron, water atoms with the icosahedron, and the cosmos atoms with the dodecahedron
Johannes Kepler (1571-1630) found that the five Platonic Solids could be inscribed and circumscribed by a each other, octahedron, icosahedron, dodecahedron, tetrahedron and cube. He tried to make a case that the known planets were ordered in some relationship like the Platonic Solids, but he never could make a scientific connection.
The 13 Archimedean Solids are composed of combinations of polygons and named after the Greek philosopher Archimedes (287-212). He originally described them but his work was lost, but rediscovered during the Renaissance by various artists. It was Kepler in 1619 that finally reconstructed the entire set. Archimedean solids are polyhedra with regular polygon faces. The faces may be different types but all the vertices are of identical combinations.
Some of the Archimedean solid can be derived from the Platonic Solids, but cutting the corners and getting truncated polyhedra. The solids no longer are regular (having the same polygon repeating itself) but all the vertices are identical.
Leonhard Euler, a Swiss mathematician (1707-1783), wrote 866 mathematical papers and books covering subjects from number theory, geometry, trigonometry, and calculus. The last two decades of his life he was blind. He discovered many theorems. A theorem is developed by mathematicians to prove a formula is always true. Proofs are associated with theorems that can be repeated. Euler discovered many theorems. He discovered a relationship between the number of faces, vertex points, and edges in all polyhedrons, no matter how irregular its shape. Euler could compute the sum of all of the face angles in any polyhedron without knowing any of their measures. The relationship is called Euler’s formula for polygons.
The following activity will have your students construct several polygons and then you will collect data to see if you can develop a formula. Make the given shape and then record the number of faces, the number of points (vertex), and the total measure of face angles in the data chart below. M = the total measure of all the face angles; v = the number of vertex points
To find the face angles, use the angle ruler. Remember a square face would have 4 ninety degree angles for a total of 360. A cube has 6 faces, so there would be a total of 6 x 360 = 2160 for “Total measure of face angles.”
Now look at the data that you have collected can you see a pattern that might help you get the information without really measuring. (clue: has to do with multiples of 360)
Look at the data and see if you can make a formula that would work on all polygons.
Does this formula work for a cone or a cylinder? No
Why or why not? Curve surfaces have no faces, hence have not angles to measure
Recreating the data and plotting the information and then graphing the data will help us determine how Euler came up with his formula. Use F = faces; V = vertex. Euler’s equation for determining the total measure of all the face angles is m=360(V-2) where m represents the total measure of all the face angles and V represents the number of vertex points. As you should have determined adding a vertex point in a polyhedron increases the total angle measure by 360 degrees. This is Euler’s equation.
The only restrictions on the figures were that there were no holes in them, that every face of a polyhedron had to be a polygon, and that all the faces were flat and edges straight.
Vertices + Total Measure of Face Angles / 360 +2
Is this important? Not really, but it provides students with an easy example of how formulas are developed. Collecting data and then looking for a relationship.